| Abstract: |
| In this talk,
we study the positive solutions to the following singular and non local elliptic problem posed in a bounded and smooth domain $\Omega\subset \R^N$, $N> 2s$:
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\begin{eqnarray*}
(P_\lambda)\left\{\begin{array}{lll}
&(-\Delta)^s u=\lambda(K(x)u^{-\delta}+f(u))\mbox{ in }\Omega\
&u>0 \mbox{ in }\Omega\
& u\equiv\, 0\mbox{ in }\R^N\backslash\Omega.
\end{array}\right.
\end{eqnarray*}
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Here $00$ and $f\,:\, \R^+\to\R^+$ is a positive $C^2$ function. $K\,:\, \Omega\to \R^+$ is a H\older continuous function in $\Omega$ which behave as ${\rm dist}(x,\partial\Omega)^{-\beta}$ near the boundary with $0\leq \beta0$ and for $\lambda>$ small enough, we prove the existence of solutions to $(P_\lambda)$.
Next, for a suitable range of values of $\delta$, we show the existence of an unbounded connected branch of solutions to $(P_\lambda)$ emanating from the trivial solution at $\lambda=0$. For a certain class of nonlinearities $f$, we derive a global multiplicity result that extends results proved in \cite{peral-al}. To establish the results, we prove new properties which are of independent interest and deal with the behavior and H\older regularity of solutions to $(P_\lambda)$. |
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